Problem: Simplify the following expression: $y = \dfrac{5x^2- 13x+6}{x - 2}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(6)} &=& 30 \\ {a} + {b} &=& &=& {-13} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $30$ and add them together. The factors that add up to ${-13}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${-10}$ $ \begin{eqnarray} {ab} &=& ({-3})({-10}) &=& 30 \\ {a} + {b} &=& {-3} + {-10} &=& -13 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({5}x^2 {-3}x) + ({-10}x +{6}) $ Factor out the common factors: $ x(5x - 3) - 2(5x - 3)$ Now factor out $(5x - 3)$ $ (5x - 3)(x - 2)$ The original expression can therefore be written: $ \dfrac{(5x - 3)(x - 2)}{x - 2}$ We are dividing by $x - 2$ , so $x - 2 \neq 0$ Therefore, $x \neq 2$ This leaves us with $5x - 3; x \neq 2$.